The length of a space curve described by the vector-valued function
r(t), 
, is
in exact analogy to the arc-length of a curve in the plane. In fact, both the planar and three-d cases can be subsumed by the formula
The arc length function s(t) is given by
Using the Fundamental Theorem of Calculus,
Recall that the unit tangent vector to a curve parameterized by t is
The unit normal vector N(t) to a curve is
The unit binormal vector B(t) is
The curvature of a curve is defined as
and, in particular,
where T is the unit tangent vector of the vector function r(t) (the first definition is parameter-free). Alternatively,
In the case of the plane, this reduces to
